Numbers, Mnemonism, Savantism, Oh My

Well, this is a bit disconcerting….

NPR recently ran an interview with author/autistic savant Daniel Tammet:

http://www.npr.org/2013/08/11/206660281/the-beauty-and-calm-of-thinking-in-numbers

I've enjoyed all three of Tammet's books. My favorite was his second one, "Embracing The Wide Sky," while the current one, "Thinking In Numbers," is a bit less consistent (from essay to essay), though still pleasant... couple of reviews here:

http://tinyurl.com/8llkr35
http://tinyurl.com/myz48jw

When I first became aware of Tammet, years ago, he struck me as a bit peculiar, especially his ability to introspectively and articulately describe his own mental processes (unusual for savants); he had an almost 'Uri Geller' style about himself, but, as he was studied by psychological experts in the field, I fully bought into his storyline. He has traveled widely and appeared extensively on television promoting his books, and there are plenty of examples of Tammet on YouTube as well:

 http://tinyurl.com/lx9oby6
(and even more of course on Google about him: http://tinyurl.com/mteo7sc )

...I've never read Joshua Foer's bestselling "Moonwalking With Einstein," but have always seen excellent reviews of it. So I was surprised to read in the comments to the above NPR piece that apparently in that volume Foer expressed doubt about Tammet's genuineness, believing he may just be a highly skilled mnemonist (memory expert) who passes himself off as a synaesthete and savant (turns out Tammet's real name is Daniel Corney though he had it legally changed over a decade ago, and Foer marshals evidence that Tammet isn't always candid about how he succeeds at what he does).

It's hard to know where the truth lies (especially since 'savantism' itself can be a bit hard to define), but the more I surfed around the Web the more suspicious Daniel's talents (or his personal portrayal of them) seemed. Even though many find Foer's skepticism unconvincing and continue to defend Tammet, others, with knowledge of memory training, do not. One of the most lengthy discussions comes at this forum site:

http://mnemotechnics.org/forums/daniel-tammet-840.html

The above has 6 pages (or 163 responses) of comments to sift through about Daniel Tammet (I haven't read them all myself, but they're not very encouraging for the authenticity of the standard Tammet-savant view). It's enough to make one a tad more leery of those neuroscience "experts" who study and report on savants (a bit reminiscent of scientists who studied and were taken in by Uri Geller decades ago) -- is the reality ever as incredible as portrayed by all the hype?

Still, savantism, including extraordinary mathematical abilities, remains a fascinating subject area, but perhaps one where a focus on Daniel Tammet is not entirely appropriate (of course the tricks of mnemonists are an interesting topic in their own right, just a different topic). Perhaps tellingly, Tammet himself, has long voiced a belief that his talents are not so rare or extraordinary, but rather are a function common to human brains, just not readily accessible to most people... that would indeed make sense if Foer's contention of extensive mnemonist training was the underlying mechanism.

Anyway, sorry if this is all old news for some of you, and you were already aware of the so-called "debunking" of Tammet; it was news to me that, given my occasional interest in savantism, seemed important to share, especially since Tammet is currently getting attention here in the States where his "Thinking In Numbers" volume was only recently widely distributed (it was available through the UK a year ago). I still enjoy the volume... but with at least a bit more of a grain of salt. And I should add, Tammet remains an interesting fellow, whether it be as a true savant, OR as someone who has successfully pulled-the-wool over the eyes of trained specialists (getting a lot of free travel and attention in the process)!


...As long as I'm mentioning books, this might be a good time to point out that Edward Frenkel's "Love and Math: The Heart of Hidden Reality" is due for release in about a month, and I suspect it may possess some of the same deep, haunting joy of math that Tammet expresses, but coming from a true mathematician. From the Amazon description:

"In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we’ve never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space."

Some more about Frenkel here:

http://math.berkeley.edu/~frenkel/Frenkel-Love-for-Math.pdf

New Math Book, Full of Mistakes

…and delightfully so!!

-- Review of "Magnificent Mistakes In Mathematics" -- by Alfred Posamentier and Ingmar Lehmann

Any volume from Alfred Posamentier is to be looked forward to, and the latest one, "Magnificent Mistakes In Mathematics" is no exception (written again with Ingmar Lehmann); a fairly quick and entertaining read for typical math buffs, with a focus not often found in math books… on famous math errors.

The book endeavors to demonstrate that even the precise, empirical field of mathematics has its share of mistakes made by prominent, knowledgeable practitioners in the road to progress. And they note that the very need to examine and explain math 'flaws' is a good thing, often leading to whole new ideas/concepts.
 
The book starts right off putting the reader at ease by highlighting "noteworthy mistakes by famous mathematicians," including such accomplished figures as Pythagorus, Galileo, Fermat, Leibniz, Euler, Poincare, and several others. It's as if to say 'if the remarkable Euler blundered why should YOU dread making math mistakes.'  Many of these errors are well-known, but still interesting, or in some instances even humorous (for example a blackboard mistake by Enrico Fermi that ended up saved for posterity on an Italian postage stamp). Interestingly, at the end of the chapter it is mentioned that the also extraordinary Carl Friedrich Gauss was not known to have made mistakes in his published material.

Chapter 2 embarks on the progressive journey through mathematics with a look at mistakes in arithmetic. This may be the least interesting, or most mundane of the chapters, and is followed by chapters that delve, in order, into algebra, geometry, and probability and statistics errors; a seeming natural progression from the more abstract to the more applied or real-life-type examples. 
Geometry mistakes, of course are often a matter of misperception or interpretation (moreso perhaps than algebra, where mis-computation may be more frequent). In the geometry realm I was very surprised that the volume leaves out one of my very favorite 'mistakes,' which goes around the internet from time-to-time, demonstrating that pi=4: http://www.bestwtf.com/2010/11/explaining-why-pi-is-4.html The sort of error involved, "misleading limits," is documented in the book with other classic examples, but still, the circle-inscribed-in-a-square paradox is so good it ought not be missed.

Of course probability and statistics are perhaps the cause of more slippery mathematical mistakes than any other area, even among professional mathematicians. It is often famously told that even the great Paul Erdös initially had difficulty with the 'Monty Hall problem,' so it is a good chapter to end the book with. Many deceptive probability conundrums of recent times are now pretty much classic, and continue to elicit great debate when heard for the first time.

There are LOTS of types of examples used through the book, demonstrating how varied the sources of math mistakes can be. Having said that, there is also sometimes redundancy to the many examples employed for any one sort of error; but using multiple examples to make a point is not necessarily a bad thing.

For the professional or broadly-read mathematician this won't likely be a highly substantive or weighty math read. The bulk of examples in the volume are well-known, but what is new is bringing them altogether under one cover-to-cover format… an entire book focused contrarily not on what math does right, but on where it may go wrong. I think this somewhat unique approach makes the volume a worthwhile, entertaining addition to one's math bookshelf, and it may be particularly useful to secondary school teachers, providing a lot of grist for instructive, thoughtful examples in the classroom. As the authors repeatedly note, there is a LOT to learn from mathematical mistakes.

In short, a thumbs-up for this volume! Posamentier seems to produce a new book almost every year, and each one simply leaves me wondering what will he come up with next.

By the way, if you missed it, Posamentier did a great "Inspired By Math" podcast interview with Sol Lederman late last year here:

http://wildaboutmath.com/2012/12/16/alfred-posamentier-inspired-by-math-13/

A Book, A Cryptographer, and a Fun Guy

I'd normally post this over at Math-Frolic, but am so past-due to get a post up here... well, here goes:


1) If you're a fan of Mark Chu-Carroll's "Good Math, Bad Math" blog (one of the oldest and most popular general math blogs on the Web) you'll be happy to know his recent book  "Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation" is available through Amazon:

http://tinyurl.com/l6b5jkm 

Here's what Amazon begins by saying about it:

"Mathematics is beautiful--and it can be fun and exciting as well as practical. Good Math is your guide to some of the most intriguing topics from two thousand years of mathematics: from Egyptian fractions to Turing machines; from the real meaning of numbers to proof trees, group symmetry, and mechanical computation. If you've ever wondered what lay beyond the proofs you struggled to complete in high school geometry, or what limits the capabilities of computer on your desk, this is the book for you."

[Haven't read it myself, but assume from Mark's blog writing, it is good.]

2) I've previously mentioned my joy with "The New York Times Book of Mathematics" anthology, and its last chapter is composed entirely of wonderful profiles of accomplished mathematicians... mathematicians fascinate me as much as mathematics itself.  So I'll reach from there again today for this link to Gina Kolata's 1994 (and still hugely interesting) portrait of Leonard Adleman, the "A" in RSA encryption:

http://tinyurl.com/mmh8agy

It starts off thusly:

"There is nothing to look at in Dr. Leonard Adleman's office at the University of Southern California, no clue that the office is even occupied. There are no pictures of his wife or of his three daughters, no cartoons or mementos -- just a computer, two chairs, a desk and a blackboard. And that is fine with Dr. Adleman. For although he is an active faculty member at the university, although he is a devoted husband and father, his is a life of the mind.
"It is a life that is nourished by deep philosophical questions and the overarching beauty of mathematics. It is a life that involves days, weeks, months of pure thought, alone in his equally barren office at his home in Northridge, Calif., 30 miles from the campus. And it is a life that has led Dr. Adleman to play a central role in some of the most surprising, and provocative, discoveries in theoretical computer science."

...and just gets better and better from there.

3) Finally, a blast from the past… if you're ~55 or over a trip down memory lane….
Re-reading an old Jeremy Bernstein volume ("Cranks, Quarks, and the Cosmos") recently, I came across his chapter on Tom Lehrer (unfortunately I can't find a free full copy of it on the internet). Lehrer was a popular, clever, irreverent singer/satirist of the '50s, '60s and 70's before he largely retired from the scene (he was, by his own admission, a huge fan of Danny Kaye, whose style he very much emulated).
During his heyday, I honestly only found his material mildly amusing (there were LOTS of satirists around in those days), and it was only years later that I discovered his background (…which made him a far more interesting figure to me!)
So for any who don't know, Lehrer entered Harvard at the age of 15 and majored in mathematics, graduating at 19; a year later he got his Masters degree (and worked on, but never completed PhD. work). He taught for awhile at Harvard, MIT, and Wellesley. Joining the Army in 1955, Lehrer worked a couple years for some outfit called… drrrrrrumroll… the NSA!

If you're unaware of Lehrer you can still find his work on YouTube:

http://tinyurl.com/m3n43zv

(He's 85 now and his own website is here:  http://www.tomlehrer.org/ )

Anyway, Chu-Carroll and Adleman may well be fun guys as well for all I know, but Lehrer took math and fun to the level of turning it into a living. Nice work if you can get it (actually, a lot of his satire wasn't math-related).

Here's one sample of Lehrer's 'mathematical' work ("Lobachevsky"), to give you his flavor:

Keith Devlin… as you may not have heard him before….

Math-Frolic Interview #15 (...not the usual fare)

"As a naturalized American, I have an immigrant's reverence for those words of our National Anthem, 'Land of the free, home of the brave.' For many of my fellow citizens born here, I fear these are just words they learned to recite in elementary school. For the fact that 56% of Americans declare that they would give away fundamental freedoms to reduce the risk of terrorist attack indicates that we may become the 'land of the enslaved, home of the scared.' "  -- Dr. Keith Devlin in Huffington Post
                                            

I'm not sure there's anyone more generous with their time and energy than Keith Devlin….
Since Edward Snowden's NSA revelations (and subsequent controversy), I noticed Dr. Devlin expressing himself on the topic (as much as one can in 140 characters!) on Twitter (@ProfKeithDevlin) more than any other mathematician/scientist I follow. The passion of his opinions intrigued me and I asked if he'd do another Math-Frolic interview, but this time just on his views about this NSA controversy -- NO MATH (my prior math interview with Keith is HERE) -- I thought he deserved more time or space than available on Twitter or even Huffington Post, where he has a piece. So read up, Keith doesn't much mince his opinions!

 ***********************************

1) The recent NSA revelations have generated a broad range of opinion across the spectrum (from outrage, to 'ho-hum, nothing new here'). Of the many math/science persons I follow on Twitter you've been among the most harshly outspoken. Can you explain a little more deeply where that sense of betrayal stems from… as a 'naturalized' citizen do you perhaps appreciate American democratic ideals even more than a lifelong born-citizen who just takes them for granted… and how much does your British background (another country with a strong democratic history) come into play? Or, is it mostly just a straightforward legal/Constitutional issue for you, unrelated to background?

Appreciating the iconic ideals of US Democracy as enshrined in the Constitution is part of my outrage at the way the US (the nation, not just the government) has allowed those ideals to slide. I certainly am under no illusions as to the many deficiencies in the US, for instance, its third-world levels of poverty and infant mortality and its medieval prison system. But those words of the Founding Fathers are one of Humanity's greatest achievements. It's surely worth remembering that the US owes its existence to the fact that those Founding Fathers were traitors. So far, everything I have seen of Edward Snowden puts him into the same camp as the Nation's founders. Certainly, his public statements and actions so far qualify him as a greater American than the current President who complicitly -- and secretly -- allowed the slide away from the founding ideals to continue.

The personal twist in my case is that, twice in my life, I've found myself as the "suspected outcast". I'll describe the first. As a young assistant professor in Germany in the 1970s, when Germany was struggling with massive student unrest and genuine internal terrorism (Bader-Meinhoff, etc.), security forces surveillance of some of my students brought me into their radar, and for a few weeks I was followed around and my mail was regularly intercepted. The fact that I was very aware of this indicated that the intention was probably to scare me as much as to find any incriminating evidence against me, and my wife and I actually regarded the whole affair with amusement. I was clearly low level, peripheral fish in their surveillance sweep, and after a few weeks they were (as far as I know) out of my life. Still, it made me realize how easy it is for a totally innocent individual to find him or herself on a government security list, simply by virtue of the people they interact with. (To this day I have no idea if any of my students were active in the political unrest of the time, or indeed if any engaged in illegal activities.)

The second time was in the UK, and contributed to my leaving my homeland for the US, but the one example I have described should be enough to indicate why I simply don't buy the frequently touted idea that "If you have done nothing wrong, you have nothing to fear." In the age of big data, just as we can easily find ourselves with a wrecked credit rating that can take years to sort out, so too we can find ourselves on a government "person of suspicion" list. In my case, I had the psychological strength to shrug it off -- albeit I did emigrate from the UK to the USA. Had I a different psychology, the ending could have been tragically different, as it was for Aaron Swartz, who was unable to sustain the inhuman persecution by US attorney Carmen Ortiz and Massachusetts assistant US attorney Stephen Heymann, who clearly viewed him as a mere pawn to advance their careers.

2) Related to the above, you actually worked for the NSA at one time in your life (contractor???). I suspect you can't say a lot about that work, but can you say, in a more general way, if that specific experience with NSA, contributes to your strong feelings? And are there any details from your own NSA experience you can tell us about which are pertinent to this ongoing story?

In the early post-9/11 era, I did work on a large, non-classified (albeit not publicized) project to improve the quality of the actionable intelligence that could be obtained from massive amounts of data. I was glad to play my small role, though when that project came to an end, I held the same view I did at the start: absent a significant HUMINT lead (human intelligence), trawling through massive amounts of data is a waste of time. There is no chance you will be able to prevent a terrorist attack. I know that a number of intelligence leaders have made statements of late that claim otherwise, but all I can say is that after working hard on the problem for five years, I reached a very different conclusion. To be sure, I do not know the computing capabilities the NSA has, but based on my understanding of the problem, without a good HUMINT lead, it won't be enough. Mathematically, the problem is known as combinatorial explosion. (With a HUMINT lead, on the other hand, you don't need to trawl the data, you just have to search for confirming evidence, starting from that one lead.)

3) Author Kurt Eichenwald wrote a book a few years ago, "500 Days," apparently divulging info similar to what Ed Snowden has revealed. Have you by any chance read this volume? He claims on Twitter (as do others) that there is nothing new in the current revelations, and that in fact some of the details, as covered by the press, are simply wrong or misleading. To those who would say, there's no real news here, and moreover, private companies (Google, Facebook etc.) snoop on individual lives FAR MORE than the Gov't., what would you say?

I've written elsewhere (Huffington Post) that the Snowden revelations were akin to Lance Armstrong's appearance on Oprah. In both cases, we learned nothing we did not already strongly suspect. But making it public knowledge, as opposed to widely believed suspicion, changes the debate. In the Armstrong case, within days, he had lost all his multi-million dollar sponsorship deals. After Snowden, the intelligence chefs could not respond to questions by saying there was no wrong doing, they had to provide actual details, in at least one case revealing a clear-cut case of perjury before Congress. Maybe heads will roll -- they should -- but maybe not. (In fairness to those involved, the nature of intelligence does put people in a difficult position with regard to being truthful. Few of us have the courage of Edward Snowden.) In any event, even ordinary citizens had a pretty good idea of what the NSA was doing, so for sure our enemies did. Statements that the Snowden revelations damaged national security are clearly absurd. The security lies in the data, not the knowledge that is exists. The only damage from the Snowden revelations is the embarrassment of people in power. (It surely cannot be international relations, except on the surface, since all the other countries harbored the same suspicions as we all did, and for sure the many countries with the technological capabilities knew for sure what we were up to!)

4) Personally, while the massive net for "metadata" concerns me, what troubles me even more (and doesn't get much coverage) is the potential for NSA individuals to target specific politicians/leaders for scrutiny and use that info for strictly partisan purposes… possessing knowledge about the personal lives of political opponents is an even greater danger to democracy than knowledge of the general citizenry. Any thoughts?

This is the real worry. Right now, we have President Obama saying "Trust me, this immense security apparatus is being used for your safety." As it happens, I am inclined to give him that trust, though in so doing I am making a leap based on no first-hand knowledge. But that's not the point. Who knows who will hold the reins in the future? It was not long ago that J Edgar Hoover was in charge of the FBI. We've had despots in positions of power before, it can happen again. When I was living and working in West Germany, I traveled occasionally to East Berlin to consult with university colleagues, and learned enough about the STASI to never want to live in a state with such a powerful and intrusive security apparatus.

5) Some people view Snowden (thus far) as a highly-intelligent, sincere, courageous, deeply-patriotic  individual, and others label him narcissistic, self-aggrandizing, delusional (some have even said, why can't he be ALL of the above!). Care to say, how you would characterize him?

I already did. I think history will portray him as a twenty-first century "Founding Father", who initiated a return to the principles by which the country was founded. Assuming, that is, that we do indeed step back from the abyss. The current attempts to discredit him are as predictable as they are transparent. His personal character actually makes little difference. He did the US a great service (that's the part history will remember) by performing a heroic act, clearly at high risk to himself. Exactly the same can be said of the Founding Fathers. Acts can endure, personalities are replaced by stories.

6) One of the interesting major disagreements is between those who say that the sort of massive "dragnet" surveillance that is going on is outright illegal and not authorized by Patriot Act measures, versus those who say there is NO "surveillance" but only the collection of large-scale metadata (which does not constitute surveillance), and only when a 'pattern' of interest is found in the data can the Gov't. then seek a court order to do further actual surveillance. I know you are interested in the uses of language and meaning, and clearly that is what we have here… Any comments?

It is clearly illegal, being against the Constitution. It's also immoral. Period.

7) Do you feel very disillusioned (as some do) by the Obama presidency over the various issues of transparency/secrecy that have arisen, or are your issues more with the intelligence community than with the White House?

We live in a democratic republic. The intelligence community do their job, and implicit in that is to collect as much information as they can. The elected government are the ones setting the limits and calling the shots. If there has been a breakdown in that line of command, it is the government that has the responsibility to put things right. If ever we were at a juncture where a president should offer real leadership, now is that time. I understand Obama would like to go down in history as another Lincoln. Now is his chance. I wonder if he has it in him.

8) Supposedly Glenn Greenwald/Guardian have several more disclosures to make from the information Snowden provided. Care to make any predictions (and I know you think that predictions, especially about the future, are difficult ;-) about what may happen over the course of say the next year? …Will Snowden be extradited and prosecuted here in the U.S.? Will the Patriot Act be revisited and revised by Congress? Will the stand politicians' take on this affair (with or against Snowden/NSA) have a major effect on the 2014 mid-term elections?….

Since I don't know what information Snowden has, I don't see how anyone can make predictions. Whatever he has clearly already exists in multiple copies, held by different people, so it will likely eventually come out. So in practical terms, the best option for the US is to simply leave Snowden alone in Hong Kong. Public interest being as it is, "the Snowden story" will soon go away -- though I hope that real reforms result. Trying to have him extradited to the US, in contrast, will not only keep the story on the front pages for months and more likely years, but if the attempt succeeds, we will have a martyr on our hands. And martyrs are dangerous. Do we want to turn Snowden into another Nelson Mandella? How do we respond if, for instance, an imprisoned Edward Snowden is awarded the Nobel Peace Prize? (Those Scandinavians have a strong sense of social justice and are not easily pressured, so that could very well happen!) Better not to go that route. There is a slew of downsides, but the only "upside" is revenge, and there is no way the US could come out with dignity and respect if we throw our immense power going after one of our own citizens so it would prove to be a hollow upside.

9) And one last crystal ball inquiry… many have contended for a long while now that in the future there simply will be NO privacy… some think current young generations have ALREADY forfeited any significant concern over privacy. I truly wonder if, a century from now, "privacy" won't be just a quaint little term in historical footnotes. You and I might not wish to live in that world, but is not the near-complete loss of privacy inevitably coming? :-(

I think that here in the US we have a choice. In 1789, a bunch of traitors to the ruling authority formulated the First and Fourth Amendments as they set the new nation on its course.  Like him or hate him, Edward Snowden has put the questions of public information and personal privacy on the table once again. As a result, we have an opportunity to correct our course. Because of the Founding Fathers, we are currently able to debate this issue freely and openly. If we don't live up to those two-hundred-years ideals now, that great episode of human society (great for all its flaws, which lie in the execution, not the ideals) will have come to an end. We will be the "Land of the enslaved, home of the scared."

***********************************

....I don't completely agree with everything Keith says here, but I surely love the man's passion... as he demonstrates in everything he takes an interest in or speaks about. And further, as someone who has experienced unwarranted governmental suspicion/surveillance elsewhere -- albeit by his admission short term and low level -- his views deserve close attention. THANKS again for taking the time to respond Dr. Devlin.

I'll close out (...for some comic relief) with this "Good Will Hunting" scene that I've already used over at Math-Frolic, and most of you have likely seen:

Waxing Platonic…

The Platonic divide in math....

Ramanujan

The book I've highlighted recently, "The New York Times Book of Mathematics," ends with a chapter of readings on various notable mathematicians… Erdos, Ramanujan, Conway, Gödel, Wiles, etc. I suspect most (if not all) of the brilliant figures profiled were/are Platonists (mathematics is discovered, not merely created). Yet many other recent math figures (Reuben Hersh, William Byers, Keith Devlin, Jim Holt, and more) have forcefully argued that mathematics is indeed a mental creation that might even differ considerably in a different Universe than ours -- indeed some almost seem to find the notion of mathematical Platonism so wrong-headed as to be silly (while Martin Gardner found the non-Platonist view almost silly). And occasionally such writers cause me to sway toward their non-Platonist stance though I always seem to float back toward Platonism.

One thing that so many of the greatest, most productive mathematicians seem to share is an uncanny, almost inexplicable ability to tap into a realm of intuition or mental landscape not readily accessible to most of us. Ramanujan is certainly the unparalleled, most inexplicable, example of this; producing amazing mathematical results that are still today being explored and proven. Reading James Gleick's portrait of Ramanujan in the Times volume it really hit me… was Ramanujan, who routinely produced such results/theorems without ever showing the steps that led to the outcome, in direct access of the "Platonic realm?" He himself claimed his insights came in dreams and trances directly from the Indian Goddess Namagiri... Who are we to argue (and where did she reside)?!!

In many ways, Ramanujan's extraordinary talents are reminiscent of the incredible abilities of various mathematical savants and prodigies who usually can't explain how they do what they do. Their brains seem clearly to operate, or even be wired, differently from those of 'ordinary' people.

My point in all this is simply that such rare, yet nonetheless real, individuals DO give an appearance of tapping into a realm… call it perhaps the Platonist realm… that the rest of us lack ready access to, where numbers and math really DO exist apart from our day-to-day world.  Naysaying non-Platonists will simply argue that however Ramanujan and the rest gain their special knowledge, it ultimately still arises via the firing of neurons within a physical human brain situated between two ears… i.e. it is still a human creation. I can't prove that reductive view wrong, but the notion that there are worlds out there that only some of us can easily tap into, and only some of the time, through means we don't even comprehend… is so much more appealing! As Shakespeare put it long long ago, “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.”

I think Martin Gardner might well relate to this idea too… For all his empirical skepticism, Gardner also described himself as a "Mysterian," a philosophical view which holds that ultimately consciousness cannot be explained by any human brain. In the famous words of computer scientist Emerson Pugh, "If the human brain were so simple that we could understand it, than we would be so simple that we couldn't."  Is it possible that humans are able to draw upon a Platonic world, and can recognize 'consciousness,' yet perhaps never, with our limited minds, fully grasp either? Does the 'Platonic world' exist, but like the Continuum Hypothesis, fall into a nether land of things that simply can't be proved true or false by human logic?
Speaking of certain mathematical proofs, Paul Erdos would famously say, "This one is from The Book!" I'm not so sure he was speaking in metaphor... perhaps The Book, in some (Platonic) manifestation, exists. Is the alluring beauty of math only in our heads, or is it an integral part of all creation? MIT physicist Max Tegmark has argued for some time now that the entire physical universe, as we perceive it, is nothing more than mathematics, or a mathematical structure (called the MUH, or "mathematical universe hypothesis").

Anyway, read Gleick's beautiful 1987 portrait of Ramanujan and just imagine the Indian mystic-mathematician dreaming and dipping into a realm where numbers are as 'real' as rocks and chairs are to most of us:

http://www.nytimes.com/1987/07/14/science/an-isolated-genius-is-given-his-due.html?

a couple of brief lines from therein:

" 'When he [Ramanujan] pulled extraordinary objects out of the air, they weren't just curiosities but they were the right things,' said Jonathan M. Borwein of Dalhousie University in Halifax, Nova Scotia...
" 'He seems to have functioned in a way unlike anybody else we know of,' Dr. Borwein said. 'He had such a feel for things that they just flowed out of his brain. Perhaps he didn't see them in any way that's translatable.' "

It's probably also worth noting that the very first entry in the entire NY Times anthology is a 1998 George Johnson piece also addressing the subject of Platonism:

"Useful Invention or Absolute Truth: What Is Math?" by George Johnson

At the end of the piece, Johnson cites a 1995 book, "Conversations on Mind, Matter and Mathematics" that covered a debate between French mathematician Alain Connes and French neurobiologist Jean-Pierre Changeux over the subject of math Platonism. An interesting and rich review of that book here (even makes brief reference to Ramanujan):
 
http://www.timeshighereducation.co.uk/161513.article 

Connes and Changeux didn't resolve the debate... and we won't here... but still, nourishing food-for-thought.

Math via The New York Times

Non-technical math anthologies are rare critters… when one comes along my instinct is to pounce on it. "The New York Times Book of Mathematics," edited by Gina Kolata, was worth the pounce!

This volume covers a wide and interesting array of topics. Here are the 7 chapter headings (though they don't fully hint at the range of material touched on):

1) What Is Mathematics?

2) Statistics, Coincidences and Surprising Facts
 
3) Famous Problems, Solved and As Yet Unsolved
 
4) Chaos, Catastrophe and Randomness
 
5) Cryptography and the Emergence of Truly Unbreakable Codes
 
6) Computers Enter the World of Mathematics
 
7) Mathematicians and Their World

That should give you a sense of the breadth of topics on display here. The pieces are vibrant, terse treatments (no doubt only intended to fit within a certain column length). The writing is so good that the pithiness leaves one reaching the end of most pieces wanting more... just one more page pl-e-e-ease.

I think Gina Kolata sets the tone and 'feel' of this engaging volume very aptly when she writes in her Introduction:

"A mathematician once dismissed the very idea that people outside his circle could ever understand the true essence of the field. Mathematics is an art form, like music or painting. Translating math into the English language, he said, is harder than translating Chinese poetry. The beauty is lost, the elegance, and a proof that is a thing of ineffable iridescence becomes reduced to a baffling or mundane-sounding bottom line....

"But even if the rest of us cannot appreciate mathematics as an art form, are we really shut out? Articles in the New York Times may not give the details of proofs, but they reveal a rich world that can be exciting, surprising, and can even tug at the heartstrings."

Yet several reviews I've seen of the volume are rather ho-hum about it, but these are usually from professional mathematicians -- for the working mathematician there may not be that much here to excite -- although I think any math lover will find at least a few pieces that strike a chord. But for lay folks with an interest in math (my core readership!!) this may be the BEST anthology I've ever come across! There is no technical material or equations to weigh down your enjoyment, nor slow your consumption. It is all about math and mathematicians... without doing math.

One downside is that because this is limited to NY Times' writers, many excellent popular math writers are absent.  Indeed, I'd normally be skeptical of an "anthology" that was restricted to the number of writers this one is -- it is very heavy on pieces from Gina Kolata and James Gleick -- but these writers are SO good at their craft that skepticism quickly fades away. While Kolata and Gleick's pieces are perhaps the best, there are numerous fine contributions as well from George Johnson, John Markoff, Dennis Overbye, and others. Oddly, there are no entries from Steven Strogatz here (author of some of the most popular math pieces the Times has carried in recent years), but perhaps his offerings simply didn't make the 2010 cutoff for the volume. The one other thing that may be missing from the collection, it seems to me, are more articles which relate math to the other sciences, particularly physics and biology (I believe the Times has run several such pieces).

Most of the entries come from the last 3 decades or so, but some go back as far as the late 1800s. I wasn't particularly enamored of several of the older entries that were probably included more for the sense of history or progression they illustrate than for the math covered. Still, overall the mix is appealing.
 
Chapter 5, focusing on cryptography, would have been interesting in its own right, but became even more-so, in light of current events, as almost every article makes mention of the NSA and its relationship with mathematicians (by most accounts, by the way, NSA is the largest employer of mathematicians in the world). But there isn't a bad chapter in the volume.

In short, I love this compendium, even more than I expected to. If it wasn't such a thick, heavy volume I would almost recommend it, at this time of year, as a 'beach-read'… for the mathematically-inclined. In the distressed world of print journalism, the NY Times has been cutting back on science journalism, so it is wonderful to have this hard-copy of delicious math-related essays to keep on one's shelf as a permanent source of popular math writing stretching across decades. Hats off to Ms. Kolata on a job well-done!

Flipped Classrooms, MOOCs, and Having a Blast

Timing is everything....

Well, this was great… I was planning to write a post musing a bit more about math education in regards to both "flipped classrooms" and MOOCs… but then discovered Keith Devlin has just put up a new (longish) post on his MOOC blog saying most of what I wanted to say, and with more authority than I could say it. So please read it:

http://mooctalk.org/2013/06/03/the-mooc-will-soon-die-long-live-the-moor/

Do note that I think his title may be a bit misleading so follow carefully all he has to say. I was afraid his long lapse in blogposts might mean that the 2nd rendition of his 'mathematical thinking' MOOC hadn't proceeded well (though his insanely busy schedule could also account for it), and luckily it doesn't sound like that was the case… though he does still write with caution about MOOCs, and will have more to say in the future about this last go-around.

Here are a few of the most trenchant comments he makes (I've added some emphasis):

"the vast majority of people under twenty now interact far more using social media than in person.
We could, of course, spend (I would say “waste”) our time debating whether or not this transition from physical space to cyberspace is a good thing. Personally, however, I think it is more productive to take steps to make sure it is – or at least ends up – a good thing. That means we need to take good education online, and we need to do so for the same reason that it’s important to embed good learning into video games…
"The media of any age are the ones through which we must pass on our culture and our cumulative learning."

"Something else that digital technologies and the Web make possible is rapid iteration guided by huge amounts of user feedback data – data obtained with great ease in almost real time."

The one place where I think Keith sounds a little too negative is when he writes:

"Experimentation and rapid prototyping are fine in their place, but only when we all have more experience with them and have hard evidence of their efficacy (assuming they have such), should we start to think about giving them any critical significance in an educational system which (when executed properly) has served humankind well for several hundred years. Anyone who claims otherwise is probably trying to sell you something."

Actually, I think "experimentation and rapid prototyping" may now be an integral part of our quickly evolving world and education system… more than ever before change can happen with such speed that we may try 4 failed experiments and still succeed at #5 in an acceptable/practical amount of time (even before the "hard evidence of efficacy" is fully in or agreed upon. Just the speed with which the MOOC movement has grown is a testament to that, and as Keith implies, the time is ripe for us to "make sure" they [MOOCs] work in some form.

So much for MOOCs…
What actually got me thinking again about education was a recent Twitter tweet that led me to this blog I was previously unfamiliar with:

http://flippingwithkirch.blogspot.co.uk/

Despite the uk URL appendage it's from a California high school math teacher (Crystal Kirch) focused on the "flipped classroom" concept. Just scanning over it, it looks interesting and impressive to me, but as someone not in the loop of secondary education I don't want to assume too much. What definitely caught my attention though (and those of you in secondary education likely already knew this) was the sheer number of other blogs with a similar focus on flipped instruction (as well as a network of teachers with this interest) that Mrs. Kirch links to. The "flipped classroom" has been around long enough that LOTS of teachers are trying it, tweaking it, playing/experimenting with it, blogging about it, and just generally sharing their experiences (good and bad) with their peers. What a great collaborative endeavor!!… and not brought on by some agency-directed-commissioned group-on-high, but by the spontaneous interest of those who share similar goals. Again, before the internet this sort of rapid cross-communication effort wasn't possible.

The term "flipped classroom" came about, so far as I'm aware, from early uses of Khan Academy videos (and Khan Academy still has many vocal critics), but of course there are now MANY internet resources available to choose from, and Khan itself constantly evolves. (Some have noted that the 'idea' of the flipped classroom, though not the term itself, actually long precedes Khan Academy.)

It is fascinating to me how both "flipped classrooms" and MOOCs, which in some ways share little in common, and operate on different levels of education, have simultaneously sprouted up like mushrooms in the cyber landscape, both controversial and rapidly-evolving, yet giving tremendous promise.

As Keith writes so aptly at the end:

"Those of us in education are fortunate to be living in a time where there is so much potential for change. The last time anything happened on this scale in the world of education was the invention of the printing press in the Fifteenth Century. As you can probably tell, I am having a blast."

And some of us are just having a blast... watching those of you who are in the trenches having a blast.
To Keith, and Mrs. Kirch, and all others doing the nitty-gritty work that will shape the education of future generations... THANK YOU!

Vickie Kearn... She Reads 'em Before You Ever Hear of 'em

 Math-Frolic Interview #14

"Appreciating the power of math and what it has and can do for us is really important. It isn’t just a lot of numbers, it is about people and applications and improving everything we care about." -- Vickie Kearn


You're likely not as familiar with the name 'Vickie Kearn' as most of the other names I've interviewed here... but it's a joy for me to bring her forth from behind the professional curtain where she hangs out. As an editor for Princeton University Press (a favorite of mine) she shepherds a great many of the books and authors we come to love, to our bookstores for us to read. It's fun to gain a better sense of how that whole behind-the-scenes process works. Read on (I've emphasized a few bits with bold):

********************************

1) To start, can you tell readers a little about your background or anything else pertinent to your math interest and your editorial position with Princeton University Press?

I have always loved math and had teachers who encouraged this love. When I was 10 my parents moved to Venezuela so when I was 16 I had to come back to the States to go to boarding school since there were no English schools where we lived. I went to a very small school and had the same math teacher (Elsie Nunn) for three years. She was wonderful and we had math club every day. Now you might think that was a bit much but she was so exciting and told terrific stories about the people behind the math. She could do all sorts of things with simple tools like sting and cardboard. I remember that she was double jointed and could draw a perfect circle.

When I went to college the women took classes on one side of the lake and the men on the other. Lucky for me, the advanced math classes were all taught on the men’s side of the lake. I went to a Baptist school and men and women could only talk to one another on certain days of the week. Because of my math connection, I got to talk to them every day of the week. Perhaps not a reason to major in math, but a really neat benefit.

After college I taught school for eight years (elementary and middle school math) and then moved to New York City to begin a career in publishing. I had pretty much had it with the books I was given to use in my classrooms and thought that I could make a difference if I could find a way to get more interesting and useful books published, especially in math. I started my career at Academic Press as a developmental editor. I had to read all of the math textbooks, work all the problems (to make sure all of the needed information was there and that the problems could actually be worked) and write the solutions manuals. After three years I did not think I could work out one more calculus problem and really wanted to get on with finding those books I so sought. I moved to Marcel Dekker where I was an acquiring editor. I actually got to look for authors who could write the next great math text. I didn’t know if when I was hired as the math editor that I would end up also working on statistics, electrical engineering, quality control, and food science. Although I got a lot of experience, I was not making much progress with my math hunt. I then went to the Society for Industrial and Applied Mathematics which was heaven because it was all math. I worked on books and journals as well as conferences and membership drives. This was exactly what I was looking for.  This was a great job but I found that there were some titles that I longed to find but which were not good fits for the society. I wanted to reach out to other disciplines to show how math could be used not just in the sciences but also the humanities and the social sciences. I wanted to bring more math to general readers—the math lovers and the math haters and math phobics. It was difficult for a math society to reach all of these different audiences without publishing books in all of these areas.

Princeton University Press has been a perfect fit for me. We publish in almost every discipline you can imagine. We have sales reps who visit bookstores and publicists who visit major media outlets for print, TV, and radio. The Press and our editorial board are willing to try new things like books of puzzles and graphic novels.

2) Prior to this point, I've been interviewing individuals who are direct math communicators, bloggers and/or authors. You're sort of a layer back as a gatekeeper of the very sorts of other folks I normally interview. That makes it interesting, because many readers won't know your name and yet you are probably more personally familiar, than those readers, with the very names they are so familiar with! Can you say a little of what it's like to work with distinguished math authors as you mentor their idea or first draft from proposal to publication? And do you build personal friendships with many of these writers as you collaborate with them over time, or is it more of a strictly arms-length business relationship?

Many of my closest friends are mathematicians.  In a way I have grown up with them and my son is the same age of many of their children. In addition to our love of math and great books, we share this common bond of raising kids, sending them to college, and watching them find their way. When I began my career in publishing in 1977 I did not know anyone. There was a lot more competition than there is now and there were many seasoned editors who had built a core of authors who always published with them. I decided that the best thing to do was to contact the people at the top of their careers (all the big prize winners) and ask them about their brightest students. These are the people I contacted and talked to about what books they needed or would have been helpful when they were studying math.  Since these were the rising stars they were soon in a position to write books and they remembered me when they were considering a publisher for their book. Now they are the prize winners and I am still publishing them and they are now recommending their students. Every book I have published has been special in one way or another. I have had the privilege to meet the most honored and famous mathematicians of our time. I also have had the honor to meet some of the greatest teachers around the world. You don’t have to win a lot of prizes to write a good book. You do have to be creative and be passionate about your subject. The trick to publishing great books is finding these people. Sometimes an author comes to me with a completed manuscript that is almost ready to go. The books that are the most fun, however, are the ones that we design together from the seed of an idea to a finished product that is widely read. This can take several years which is plenty of time to establish a lasting friendship.

 [....Sounds like a dream job!! ;-)]

3) Princeton University Press puts out some of the most consistently excellent, interesting, well-designed math books of any publisher! So I'm curious how that selection process works so successfully. Can you describe a little of how things proceed from the time a writer approaches PUP to the time a book is accepted for publication and finally produced, and what is your role along the way?

Princeton University Press cares a great deal about its authors and the books it publishes. Each book is carefully selected to ensure that it is accurate, fits a specific audience and is pleasant to look at and read. Sometimes authors come to me with an idea and sometimes I think of a topic I think will be great for a book and seek an author who would be perfect to write it. This can take a long time so you have to be patient. I have given up on a topic at times when I can’t find the right author. The first step in our process is to put together a proposal for a book. I then present it to my colleagues who help me decide if the topic fits our list and if we can promote the book effectively. If so, I have the proposal reviewed and if the readers are positive, we offer a contract. We might work on the development of the project over many years or it might come together quickly. Once the final manuscript is complete, it is sent out for a final review. If the book is for the general reader or an undergraduate textbook, I read though it as well and give the author advice on changes to consider. If the reader reports suggest more work, then the author revises and we send it back to the readers. If the suggestions are minor, I take the book to our editorial board for final approval. The board consists of five Princeton University professors across all disciplines who approve books for publication based on my recommendation and those of the readers. They ensure that the book reflects the mission of the University as well as the Press.

The production process is a careful one. We copyedit all of our books and redraw art where necessary. We have designers who look at the book to make sure that the manuscript will be laid out it the most user friendly way. They also are responsible for designing a cover that is attractive and reflects the content of the book. During production, our publicity, marketing, and sales departments are all preparing materials and contacting people to make sure that our newly published books will be as noticeable as possible and will get into the hands of readers.  During this time I am solving any problems that arise and working on getting endorsements which will go on the cover of the book. I no longer have to write solution’s manuals but I make sure that the authors are. I also help authors come up with ideas for ancillary material that they might want to put on the webpage for their book. We are also developing Facebook and Twitter accounts for each book during this time.

[Very interesting to hear about the whole process! Needless to say I think PUP achieves its goals well -- your math books are always very readable, informative, AND very attractive to look at!]

4) Roughly speaking, of proposals you get for math fare, what percentage might PUP generally end up publishing? And is it possible to generalize about what the most common reason for rejecting a proposal is?

The sciences are different from the humanities and social sciences where it is imperative to write a book or two in order to get tenure. The editors in these areas are deluged with proposals. In math, the opposite is true. They are writing and publishing research papers to get tenure. Most of the proposals I get are from direct recommendations from someone I know, an author I have already published, or someone I have approached so I don’t get a lot of unsolicited proposals. I do get a few and look at each one carefully before deciding what to do. Many are rejected for various reasons and others get published. The most common reason for rejecting a proposal is that it is totally wacky or not prepared properly. Sending an editor a proposal with hand drawn figures and no coherent description of the book or who you are writing for is a good sign the book will not be worth publishing. Just as you would never think about sending in a resume that is loaded with typos for a job application, you should check your proposal carefully to make sure it states what the book is about, why it is important, who it is for, what the reader will gain from reading the book,  and what the competition includes. Oh yes, and check for typos!

I publish a very small percentage of the unsolicited proposal I receive. However, there is a very high publication rate of those that are recommended to me or that I go looking for.

5) Obviously, any book you choose to publish, you believe is well-done and will have an audience, but are there any examples of Princeton math books that especially surprised you with the volume of their sales?

As I said, each book I work on is special in some way. It may be that it sells only 600 copies but the readers use the information inside to solve some great problem or advance a new area of math in some way. Others might sell tens of thousands and get high school students excited about math. All are important in my mind. Some books get great reviews and just don’t live up to expectations. Others get little notice in the media but find their way and outsell our expectations.

6) On the other side of the coin, have you ever been involved in rejecting a book for Princeton, only to see it become a major seller for another publisher, and thought, 'ohhh man, why did we let that one slip away!'…?

On several occasions I have rejected books that I know will sell well but which didn’t meet our mission in one way or another. Every publisher wants their books to sell as many copies as possible but not at the risk of getting negative reviews and possibly damaging a reputation that has long been established. I actually can’t think of a book that I was sorry I rejected for these reasons.

7) Can you tell us anything about some of the math titles/topics/authors we have to look forward to coming down the pike shortly? 

I just presented my fall 2013 list to our sales reps and there are some great books on that list. They include:

Undiluted Hocus Pocus: The Autobiography of Martin Gardner -- This is one of the last things that he wrote before he died. He was a very private person and even his closest friends have learned a lot from reading the manuscript.   

Beautiful Geometry by Eli Maor and Eugen Jost is an illustrated guide to some of the major ideas in geometry. It includes proofs, history and art designed just for this book.  

Wizards, Aliens, and Starships by Charles Adler is all about the math and physics in fantasy and science fiction. Which cool things could actually happen and which are impossible?  

Will You be Alive Ten Years From Now is a book of probability puzzlers by Paul Nahin who is a perennial favorite. He has published many books with us and has a very loyal following. 

 [Ohhh Wow, these sound FANTASTIC!!! And an autobiography from Martin Gardner... I'm almost drooling over the keyboard thinking about it... who knew there would be yet more Martin to enjoy 3 years after his demise. I'm not familiar with Adler, but Maor and Nahin are other favorites. THANKS so much for letting us know about these ahead-of-time!]

8) When you're not editing math books, what are some of your other main interests/hobbies/activities?  

I like to tutor kids who are struggling with math and I volunteer for a pet rescue. I also like to read and solve logic problems.

9) Any parting words, not covered above, you'd want to pass along to an audience of math readers and enthusiasts?

Every day I try to find someone who does not like math (or thinks they don’t) or thinks it is hard and convince them that math is fun and is not really that hard (at least on some level). Appreciating the power of math and what it has and can do for us is really important. It isn’t just a lot of numbers, it is about people and applications and improving everything we care about. From sports to medicine to ensuring we are safe, math plays a large part. If every reader could convert another person every day, we soon would have a hard time finding people who don’t like math.

...A great thought to end with!

THANKS, Vickie, for giving us an inside look at how the books we enjoy so much end up in our hands.

*******************************
And if you want to hear her voice, Vickie was also interviewed last year as part of Sol Lederman's podcast series here:

http://wildaboutmath.com/2012/06/03/vickie-kearn-inspired-by-math-8/

From Whence...?

"The more the universe seems comprehensible the more it also seems pointless."
-- physicist Steven Weinberg as famously quoted in Jim Holt's "Why Does the World Exist?"


I'm currently finishing Jim Holt's bestseller (and one of the NY Times' Top 10 nonfiction picks for 2012), "Why Does the World Exist?" (now out in paperback). Won't write a full review since it's more philosophy than math or science, but will recommend it to those with a philosophical bent, or who have enjoyed any of the other recent books on that most fundamental of questions, 'why is there something instead of nothing?"

The first half of Holt's book, while good, takes a little bit of time to gain traction, and the second half is especially good and invigorating. Holt keeps the sometimes deep philosophical and theoretical discussion not only accessible, but also moving along at a pace which doesn't get too bogged down with any one set of arguments or thinker. If you are someone who scoffs at the very idea of reading an entire book on a question that essentially can't be answered, then you'll want to pass on this volume, but if you enjoy seeing the wide variety of cerebral exercises major thinkers have employed to approach this basic conundrum than Holt takes you on a good ride, tossing in personal anecdotes along the way.

My favorite chapter (not too surprisingly) is chapter 10 on Platonism, where, in addition to Plato, the likes of Kurt Gödel, Roger Penrose, Max Tegmark, Bertrand Russell, and Hartry Field are among those making appearances in the debate over whether there is an independent platonic realm of mathematics (apart from human consciousness), or is mathematics merely a human construction. Both sides have very astute and brilliant proponents.

The mini-portraits of the many fascinating individuals Holt discusses or holds court with in this book are just as interesting as the ideas they put forth. John Updike fans will find a chapter toward the end (#13) with their literary hero. Heidegger, Quine, Wittgenstein, Leibniz, Richard Swinburne, David Deutsch, Adolf Grunbaum, Derek Parfit, John Leslie, Thomas Nagel, Robert Nozick, are among the intriguing panoply of players (living and dead) who are aired in these pages (many individuals who I was not previously familiar with at all).  Some of the book's philosophical discussion is a bit muddied in semantics (as could be expected), and I prefer the discussions with scientists, but Holt deftly works his way through all of it, be it religion, cosmology, or quantum mechanics. And at the end comes a moving chapter on the death of his own mother.

For some extended reviews of the book here are two (of several) from the Web:

http://bnreview.barnesandnoble.com/t5/Reviews-Essays/Why-Does-the-World-Exist/ba-p/8475

http://articles.washingtonpost.com/2013-02-08/opinions/36990295_1_wrong-question-answer-universe

Of course the volume reaches no final resolution and there is a bit of predictability in so much as certain issues keep recurring as sticking points, yet the intellectual exercise remains entertaining. Holt notes early on in the volume that the question of why the world exists is "so simple that it would occur only to a child." Maybe that explains why, despite its intractability,  we find it such an irresistible inquiry... it makes us all feel like children again in this great big farfetched universe of ours (...or, multiverse).

One last thing:
The quotation I've long-used from Bertrand Russell heading my Math-Frolic blog comes from early in his career when his optimism about mathematics and empiricism was strong. Holt's book (again the chapter on Platonism) introduced me to another quote, I was unfamiliar with, from much later in Russell's storied life. I love the quote, and change-of-heart it expresses, so much so that I've added it to the Math-Frolic blog heading, and will leave you with it here:

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal."

[NOTE: I've actually since moved the paired quotes over to the MathTango heading.]

[...This all reminds me of yet another brand new book now showing up in bookstores: not meaning to stray too far from mathematics, but computer scientist/keen-thinker Douglas Hofstadter's latest tome is, "Surfaces and Essences" (about the role of 'analogy' in human cognition), and I suspect it is must-reading for anyone interested in cognition. (Hofstadter was author of "Gödel, Escher, Bach," one of the most acclaimed works of nonfiction of the last half-century, and several works since. He also took over Martin Gardner's column at Scientific American for a few years back when Gardner retired.)]

Beholding Mathematics...

“Beauty is the first test: there is no permanent place in this world for ugly mathematics.” -- G.H. Hardy


Just a little Sunday rumination on mathematical beauty today….

Math bloggers come in a variety of flavors, sometimes with few specific overlapping interests, but what they often hold in common is an overarching desire to share math's beauty and wonder with others.
"The beauty of mathematics" is a phrase most math bloggers have probably used at some point… it rolls off our tongue (or keyboard), so obvious is the "beauty" of mathematical thought and application we see, despite its frequent lack of resonance with other lay folks.

This article about young Princeton University professor Manjul Bhargava well captures his enthrallment with the "beauty" of math:

http://www.livescience.com/28508-numbers-manjul-bhargava-nsf-bts.html

In it Bhargava says: "When you discover things about numbers, it's very beautiful. When mathematicians are thinking about their problems, we're not thinking about their various applications, but rather are pursuing beauty. That's how pure mathematicians think."

Bhargava is of Indian origin... same as the remarkable Ramanujan who also found incredible, and uncanny, beauty in mathematics, which he felt was communicated to him directly in visions from an Indian goddess. 

And most anyone who reads science on the Web has probably seen this Richard Feynman clip which goes semi-viral from time to time. In it he describes, touchingly, how he, as a scientist, is capable of perceiving beauty, just as much as one of his artist friends can do… perhaps even more-so. As he essentially says toward the end, 'How does science detract from the ability to perceive beauty… it only adds, it only ADDS!!' …and so too mathematics, one could say:



So why do people universally comprehend the notion of "beauty" when applied to art or music or fashion or even food, but not when it comes to mathematics?
Usually beauty is beheld directly through the senses, vision, hearing, taste, touch… but math is more abstract, beheld at a different cerebral level where we discern patterns or connections mentally, whether or not they are physically evident.
Certainly not all math is beautiful, and it is indeed interesting how some theorems or proofs can be viewed as elegant or beautiful, while others simply are not (EVEN though they may be equally true or valid). As Paul Erdös was fond of saying some proofs are "from the Book" (God's creation manual).

An article from Smithsonian Magazine notes "...two of the essential requirements for mathematical beauty. First, it is surprising... Second, it is simple."

Read more:
 http://www.smithsonianmag.com/science-nature/the-natural-beauty-of-math-174842751.html#ixzz2KS3JBicz

Meanwhile S.Lang, in "The Beauty of Doing Mathematics," addressing those who view mathematics as dry and dull, writes "Last time, I asked: 'What does mathematics mean to you?' And some people answered: 'The manipulation of numbers, the manipulation of structures.' And if I had asked what music means to you, would you have answered: 'The manipulation of notes?' "

Robert Krulwich, Marcus du Sautoy, and Simon Singh explore the topic in this short clip from the World Science Festival:



And below a longer piece from the same Festival (including, at the 76-minute point, comments on Platonism vs. non-Platonism in math):

http://worldsciencefestival.com/videos/mysteries_of_the_mathematical_universe

There the debate continues over whether mathematics is discovered or invented; whether it exists in some outer 'real' world or only within the processes of the human brain.
And one may reflect on and on and on about mathematical 'beauty,' but surely it at least (like all beauty), ultimately exists in the… mind… of the beholder (be that beholder Man or God).

Of P and NP

"The Golden Ticket" by Lance Fortnow


Who'd a thunk it!? …that somebody could write an engaging, fascinating account of the P vs. NP Millennium Problem for a mass audience? Moreover to have done so without ever too-technically defining either P or NP, nor introduced much of the jargon or mathematics one might expect such a treatise to require! Hats off to Scott Fortnow (a blogger at "Computational Complexity") for doing it!

P vs. NP is one of the seven famous Clay Institute Millennium Problems for which one earns a cool $1 million for simply finding a solution (… but of course "simply" is NOT the operative word! -- thus far only one of the problems, the Poincare Conjecture, has been solved… and, ironically, the solver, Grigori Perelman, refused the prize money!).

Fortnow's book has been well-praised by reviewers, and oddly it reminds me a bit of Jason Rosenhouse's "The Monty Hall Problem." Both books are uncannily the same size and shape (and about the same number of pages), but more importantly, just as Rosenhouse did a wonderful, meticulous, intricate job of explaining the Monty Hall Problem (and its many nuances) to a lay audience, Fortnow does a similar job for P vs. NP, with a book that is, surprisingly, a bit of a page-turner.

I suspect I could read Fortnow's work 3-4 more times and each time take away a little more knowledge and understanding of the depth of P vs. NP, such is the richness of the volume. The content and implications of the book crosses boundaries of computer science, philosophy, math, logic, complexity, and number theory.

For any who don't know, P problems are those that can essentially be solved relatively quickly; i.e. "efficiently" (via a computer at least), while NP problems cannot be solved in any reasonable amount of time, even with the aid of computers, even though proposed answers CAN BE checked quite quickly. And the conundrum is to discover if these two classes of problems are in fact one-and-the-same, i.e. P = NP… or, as MOST researchers believe, do P and NP problems represent distinctly separate categories. To the uninitiated it may seem like a question of hugely abstract or narrow interest, but as Fortnow notes, "Determining whether P = NP is the most important question in computer science and perhaps in all mathematics." [bold added] Again, most believe P ≠ NP, but this too is exceedingly difficult to PROVE.

The so-called "Travelling Salesman Problem" is likely the most well-known of the NP problems… given say a few hundred cities to visit, find the shortest, most efficient route for visiting them all (…might seem to some like a straightforward problem for a modern-day computer to solve… but, it ISN'T! there simply is no algorithm, nor brute force technique, to solve it in a human time-frame.
The initial chapters of the book employ made-up examples to communicate the nature of P vs. NP. Often I prefer the use of 'real-life' type examples over invented ones to get across mathematical or scientific ideas, but Fortnow's concocted versions are so good and entertaining that they serve his purposes well. Chapters 4 and 5 go into some of the history of P vs. NP… normally, a book might start off with the historical part, but I think Fortnow succeeds in drawing you into the whole subject with the introductory chapters, and only then putting forth the history which may be a little dry or tedious and not the best lead-in to the topic.

Chapter 7 touches on Gödelian thought as well as the "halting problem." Most folks are familiar with the paradoxical, self-referential sentence, "This sentence is not true." Fortnow introduces a variation, "There is no proof that this sentence is true," to help explain how Gödelian self-reference demonstrates that there exist sentences which may be known to be true, yet are unprovable (if the prior sentence is false, then there IS a proof of the sentence, but if there IS a proof that the sentence is true, then it CANNOT be false! And vice-versa IF the sentence IS true, then, by its own admission, there is NO PROOF of its truth). In the end though, Fortnow notes that "this paradoxical approach to P versus NP seems doomed to fail, at least as a direct attempt at showing P ≠ NP."

Chapter 8 is on cryptography, one of the most significant real-life areas that would be affected if P were ever shown to be equal to NP (in which case code breaking, including breaking RSA encryption, could be done with ease).

In Chapter 10 we learn about NC, 'Nick's Class' of problems, another category within complexity theory, that may or may not be equal to the NP category of problems. Such ideas may be old-hat for those well-steeped in computer science, but are a new arena for the rest of us.

One update I learned from the book is that Vinay Deolalikar's claimed 2010 proof that P ≠ NP is apparently no longer in play. Last I'd heard (but I haven't followed the story that closely) Deolalikar was still trying/hoping to patch the various objections that critics voiced concerning his 'proof.' Apparently, though there were (not surprisingly) fatal flaws he could not overcome.
In fact, according to Fortnow, "We are further away from proving P ≠ NP then we ever were. Not literally, but in the sense that there is no longer any obvious path, no known line of reasoning that could lead to a proof in the near future."
Fortnow does mention that the field of "algebraic geometry" is currently the best candidate for possibly making progress on the problem. And he does indicate toward the end that he believes eventually P will be shown to not equal NP "…but it might take twenty or two hundred or two thousand years."
This is yet another splendid book from Princeton University Press, on a topic rarely treated in book-length form for the layperson, and for most of us a valuable, mind-stretching offering.

(Princeton Univ. Press has a transcribed interview with Fortnow about his book here: http://press.princeton.edu/releases/m9937.html )

ADDENDUM: Scott Aaronson, who is far more expert than I to review the Fortnow book, now has his own very positive review up over at his blog here:

http://www.scottaaronson.com/blog/?p=1293